Given genus $0$ compact Riemann surface, is there a Hauptmodul having pole of order 1 at $\infty$?

Question

Let $\Gamma \subset SL_{2}(\mathbb{Z})$ be a discrete subgroup containing the transformation $z\mapsto z+1$. Let $\mathbb{H}$ be the upper-half plane, and let $X:= \Gamma \setminus \mathbb{H}^{*}$ be the compactified Riemann surface obtained from $\Gamma$. Assume $X$ has genus $0$. Can we always find a Hauptmodul (i.e. generator of the function field of $X$) $f$ with ord$_{\infty}(f) = -1$ ?

Answer 1

As $X$ is a compact Riemann surface of genus $0$, it is equivalent to the Riemann sphere $S$. Let $F(S), F(X)$ be the function fields of these surfaces. Then $F(S) \cong F(X)$. As $F(S)$ is generated by $g: z \mapsto z$ having pole at $\infty$ of order $1$, $F(X)$ is generated by the image $f$ of $g$ and still has an order-$1$ pole at $\infty$.